Is it ever implied anywhere exactly what happened to Doc Hudson between the events of Cars and Cars 2? At I've point one of the characters says something like "doc would've been proud of you". I assume the reason for the character's exclusion is because Paul Newman didn't get signed up.

In a Time interview with John Lasseter, he spoke about the reasons why Doc's voice wasn't recast;

Cars 2 includes an homage to Paul Newman, whose character, Doc Hudson, was the elder statesman of Radiator Springs. Newman died
shortly after making the original Cars, and now the Piston Cup,
McQueen’s big race, has been renamed the Hudson Hornet Piston Cup. Did
you cast Michael Caine as Finn McMissile to try to fill those
elder-statesman shoes?

First of all, Paul Newman and I became very close friends. I loved that man. I went to a number of races with him and shared his love of
racing. Every time I went to New York, I would have dinner with him
and Joanne. We have lost a few of our voice actors in the past and
had to find soundalikes, but I just couldn’t do that because Doc
Hudson was Paul Newman.

So, that’s why I say he was Doc Hudson and Doc Hudson was him. So,
when we started contemplating the new movie, right away we said:
“Let’s just come up with a tasteful way to somehow pay homage to the
character and to Paul Newman.” So, we came up with the idea that the
big cup that they race for is called The Piston Cup, which became
re-named The Hudson Horney Memorial Piston Cup in honour of Doc Hudson
and they turned his doctor’s office into a Doc Hudson Museum. So,
there’s just a little moment with Mater and McQueen right at the
beginning which was my kind of tribute to Paul Newman in the film.

That being said, I know of no source that confirms what the Doc actually died of in-universe (presumably simply of old age or the car equivalent of organ failure)